Procesado de Señales e Imágenes Médicas

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2025-10-16

Procesamiento de imágenes

Basic Mathematic - Element-Wise Operation

Definition

Operation involving one or more images is carried out on a pixel-bypixel basis

\[ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \oplus \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22}\end{bmatrix} = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22}\end{bmatrix} \]

\[ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \odot \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22}\end{bmatrix} = \begin{bmatrix} a_{11}.b_{11} & a_{12}.b_{12} \\ a_{21}.b_{21} & a_{22}.b_{22}\end{bmatrix} \]

Basic Mathematic - Linear Operations

Definition

Given two arbitrary constants, \(\alpha_1\) and \(\alpha_2\), and two arbitrary images \(f_1\left(x,y\right)\) and \(f_2\left(x,y\right)\), \(\varkappa\) is said to be a linear operator if:

\[ \begin{equation}\begin{split} \varkappa\left[\alpha_1 f_1\left(x,y\right) + \alpha_2 f_2\left(x,y\right)\right] & = \alpha_1 \varkappa\left[ f_1\left(x,y\right)\right] + \alpha_2 \varkappa\left[f_2\left(x,y\right)\right] \\ & = \alpha_1 g_1\left(x,y\right) + \alpha_2 g_2\left(x,y\right) \end{split}\end{equation} \]

Supose \(\alpha_1 = 5\), \(\alpha_2 = 2\), \(\varkappa = max\) and consider:

\[f_1 = \begin{bmatrix}0 & -1 \\2 & 4\end{bmatrix}\], \[f_2 = \begin{bmatrix}30 & 4 \\-2 & -3\end{bmatrix}\]

Basic Mathematic - Adding

Basic Mathematic - Adding

 

x_ray_chest = cv2.imread("../../recursos/imagenes/Presentaciones/PSIM/female-chest-x-ray.jpg")
plt.imshow(x_ray_chest, cmap="gray")
plt.show()
image_synt1 = 100*np.abs(np.random.normal(0, 1, x_ray_chest.shape))
plt.imshow(image_synt1)
plt.show()
final_image = np.uint8(x_ray_chest+image_synt1)
plt.imshow(final_image)
plt.show()

Basic Mathematic - Multiplying

 

x_ray_chest = cv2.imread("../../recursos/imagenes/Presentaciones/PSIM/female-chest-x-ray.jpg")
mask = np.uint8(np.zeros(x_ray_chest.shape))
mask[400:700, 250:600, :]=1
plt.imshow(x_ray_chest)
plt.show()
plt.imshow(255*mask)
plt.show()
plt.imshow(np.multiply(x_ray_chest,mask))
plt.show()

Basic Mathematic - Basic transformation

\[x_{\text{rescaled}} = n_{\min} + \frac{(x - \min)}{\max - \min} \, (n_{\max} - n_{\min})\]

Basic Mathematic - Pixel intensity

Exp=1.2

Exp=0.2

Exp=0.30

Exp=0.5

Original

Exp=1.1
x_ray_chest_gray = cv2.cvtColor(x_ray_chest, cv2.COLOR_BGR2GRAY)
plt.imshow(x_ray_chest_gray, cmap="gray")
plt.show()
plt.imshow(np.power(x_ray_chest_gray,1.1), cmap="gray")
plt.show()
plt.imshow(np.power(x_ray_chest_gray,1.2), cmap="gray")
plt.show()
plt.imshow(np.power(x_ray_chest_gray,0.2), cmap="gray")
plt.show()
plt.imshow(np.power(x_ray_chest_gray,0.3), cmap="gray")
plt.show()
plt.imshow(np.power(x_ray_chest_gray,0.5), cmap="gray")
plt.show()

Basic Mathematic - Pixel intensity

Taken from: Gonzalez, Rafael C., y Richard E. Woods. Digital Image Processing. New York, NY: Pearson, 2018.

Basic Mathematic - Pixel intensity

Taken from: Gonzalez, Rafael C., y Richard E. Woods. Digital Image Processing. New York, NY: Pearson, 2018.

Neighborhood operations

Taken from: Gonzalez, Rafael C., y Richard E. Woods. Digital Image Processing. New York, NY: Pearson, 2018.

Neighborhood operations

Neighborhood operations

Neighborhood operations

Neighborhood Operations

For example, suppose that the specified operation is to compute the average value of the pixels in a rectangular neighborhood of size mn × centered on \(\left(x,y\right)\). The coordinates of pixels in this region are the elements of set \(S_{xy}\).

Elderly woman image

Gray-scale image
#|

elderly = cv2.imread("../../recursos/imagenes/Presentaciones/PSIM/elderly.jpg")
plt.imshow(elderly)
elderly_gray = cv2.cvtColor(elderly, cv2.COLOR_BGR2GRAY)
plt.imshow(elderly_gray, cmap="gray")

Neighborhood operations

N = 10
kernel = np.ones((N,N),np.float32)/(N*N)
dst = cv2.filter2D(elderly_gray,-1,kernel)
plt.imshow(dst, cmap="gray")

Neighborhood operations

N=11

dst1 = cv2.medianBlur(elderly_gray, N)
plt.imshow(dst1, cmap="gray")

Neighborhood operations

Mean Filter (3×3)

Kernel

\[ K_{\text{mean}}=\frac{1}{9} \begin{bmatrix} 1&1&1\\ 1&1&1\\ 1&1&1 \end{bmatrix} \]

Purpose: Basic smoothing (box filter). Effect: Reduces high-frequency noise and blurs edges. Notes: Linear and separable if implemented as two 1D averages; normalize by kernel sum.

Gaussian Filter (σ≈1, 3×3)

Kernel

\[ K_{\text{gauss}}=\frac{1}{16} \begin{bmatrix} 1&2&1\\ 2&4&2\\ 1&2&1 \end{bmatrix} \]

Purpose: Weighted smoothing around the center. Effect: Attenuates noise while preserving edges better than the mean filter. Notes: Approximates a discrete Gaussian; often used before edge detection.

Sobel Operator (Gradient 3×3)

Kernels

\[ K_{x}= \begin{bmatrix} -1&0&1\\ -2&0&2\\ -1&0&1 \end{bmatrix},\quad K_{y}= \begin{bmatrix} -1&-2&-1\\ 0&0&0\\ 1&2&1 \end{bmatrix} \]

Purpose: Estimate gradient along (x) and (y). Effect: Highlights horizontal and vertical edges; used for magnitude (). Notes: Includes implicit smoothing; more robust to noise than Prewitt.

Prewitt Operator (Gradient 3×3)

Kernels

\[ K_{x}= \begin{bmatrix} -1&0&1\\ -1&0&1\\ -1&0&1 \end{bmatrix},\quad K_{y}= \begin{bmatrix} -1&-1&-1\\ 0&0&0\\ 1&1&1 \end{bmatrix} \]

Purpose: Estimate gradient with a simple scheme. Effect: Directional edge detection. Notes: Less smoothing than Sobel; computationally cheaper.

Laplacian (2nd Derivative)

Kernels

4-neighbor version: \[ \begin{bmatrix} 0&-1&0\\ -1&4&-1\\ 0&-1&0 \end{bmatrix} \]

8-neighbor version: \[ \begin{bmatrix} -1&-1&-1\\ -1&8&-1\\ -1&-1&-1 \end{bmatrix} \]

Purpose: Enhance abrupt transitions (zero-crossings). Effect: Isotropic edge detection; very sensitive to noise. Notes: Often preceded by smoothing (LoG/DoG). May introduce halos.

Sharpen Filter (3×3)

Kernel

\[ K_{\text{sharpen}}= \begin{bmatrix} 0&-1&0\\ -1&5&-1\\ 0&-1&0 \end{bmatrix} \]

Purpose: Reinforce high-frequency components (details). Effect: Produces a crisper image; enhances edges and textures. Notes: Equivalent to adding the original image with a high-pass component; watch for saturation and noise.

Roberts Cross Operator (2×2, Diagonals)

Kernels

\[ K_{x}= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix},\quad K_{y}= \begin{bmatrix} 0&1\\ -1&0 \end{bmatrix} \]

Purpose: Diagonal gradient (45°/135°). Effect: Detects diagonal edges quickly. Notes: Very noise-sensitive; useful in real-time or hardware systems due to small size.

Emboss (Directional Relief)

Kernel

\[ K_{\text{emboss}}= \begin{bmatrix} -2&-1&0\\ -1&1&1\\ 0&1&2 \end{bmatrix} \]

Purpose: Simulate relief under oblique lighting. Effect: Produces shadows and highlights (3D appearance). Notes: Requires recentering/normalization to avoid brightness shift.

Neighborhood operations

Taken from: Gonzalez, Rafael C., y Richard E. Woods. Digital Image Processing. New York, NY: Pearson, 2018.

Neighborhood operations

Taken from: http://datagenetics.com/blog/august32013/index.html

Edge dection

Edge dection

Edge dection

dst = cv2.Sobel(elderly_gray, cv2.CV_16S, 1, 0,  ksize=3, scale=1, delta=0, borderType= cv2.BORDER_DEFAULT)
dst1 = np.uint8(255*dst/np.max(dst))
plt.imshow(dst1, cmap="gray")

Histogram

elderly_hist = cv2.calcHist(elderly_gray, [0], None, [256], [0,256])
plt.plot(elderly_hist, color="gray")

Histogram

3 3 2 2 1 1 0 3 0 1
2 2 2 3 2 2 0 2 2 1
0 3 2 0 1 1 3 1 1 1
3 0 2 0 2 3 1 0 2 1
2 2 0 0 3 1 3 1 3 1
3 3 2 0 3 0 3 2 0 3
3 3 1 1 2 3 0 3 1 3
3 1 3 3 2 0 3 0 2 1
2 1 1 3 3 1 3 2 2 1
0 3 2 2 1 1 0 0 0 0 

elderly = cv2.imread("../../recursos/imagenes/Presentaciones/PSIM/elderly.jpg")
elderly_gray = cv2.cvtColor(elderly, cv2.COLOR_BGR2GRAY)
plt.imshow(elderly_gray, cmap="gray", vmin=0, vmax=255)
plt.show()


elderly_hist = cv2.calcHist(elderly_gray, [0], None, [256], [0,256])
plt.plot(elderly_hist, color="red")
plt.show()

cv2.calcHist(images, channels, mask, histSize, ranges)

Help Docs Opencv

Histogram

def adjust_gamma(image, gamma=1.0):
   invGamma = 1.0 / gamma
   table = np.array([((i / 255.0) ** invGamma) * 255
      for i in np.arange(0, 256)]).astype("uint8")

   return cv2.LUT(image, table)

elderly = cv2.imread("../../recursos/imagenes/Presentaciones/PSIM/elderly.jpg")
elderly_gray = cv2.cvtColor(elderly, cv2.COLOR_BGR2GRAY)
plt.imshow(elderly_gray, cmap="gray", vmin=0, vmax=255)
plt.show()
elderly_hist = cv2.calcHist(elderly_gray, [0], None, [256], [0,256])
plt.plot(elderly_hist, color="red")
plt.show()
elderly_gray_light = adjust_gamma(image=elderly_gray, gamma=2)
plt.imshow(elderly_gray_light, cmap="gray", vmin=0, vmax=255)
plt.show()
elderly_hist_light = cv2.calcHist(elderly_gray_light, [0], None, [256], [0,256])
plt.plot(elderly_hist_light, color="red")
plt.show()
elderly_gray_dark = adjust_gamma(image=elderly_gray, gamma=0.3)
plt.imshow(elderly_gray_dark, cmap="gray", vmin=0, vmax=255)
plt.show()
elderly_hist_dark = cv2.calcHist(elderly_gray_dark, [0], None, [256], [0,256])
plt.plot(elderly_hist_dark, color="red")
plt.show()
elderly_gray_lowcontrast=np.uint8(0.1*elderly_gray)+172
plt.imshow(elderly_gray_lowcontrast, cmap="gray", vmin=0, vmax=255)
plt.show()
elderly_hist_lowcontrast = cv2.calcHist(elderly_gray_lowcontrast, [0], None, [256], [0,256])
plt.plot(elderly_hist_lowcontrast, color="red")
plt.show()

Histogram Equalization

Algorithm

  1. Calculate Histogram: Calculate the histogram of the original image, showing the frequency distribution of each intensity level.
  2. Calculate Cumulative Distribution Function (CDF): Calculate the cumulative distribution function (CDF) of the histogram. The CDF represents the cumulative sum of frequencies for each intensity level.
  3. Equalization: For each pixel in the original image, calculate the new intensity value using the formula: \[New_value = (CDF(old value) * (L-1))\] where L is the number of intensity levels (e.g., 256 for an 8-bit image).
  4. Assign New Values: Assign the new intensity values calculated in step 3 to the equalized image.

Histogram Equalization

plt.imshow(elderly_gray_lowcontrast, cmap="gray", vmin=0, vmax=255)
plt.show()

plt.plot(elderly_hist_lowcontrast, color="red")
plt.show()

elderly_hist_equ = cv2.equalizeHist(elderly_gray_lowcontrast)
plt.imshow(elderly_hist_equ, cmap="gray", vmin=0, vmax=255)
plt.show()

elderly_hist_equ = cv2.calcHist(elderly_hist_equ, [0], None, [256], [0,256])
plt.plot(elderly_hist_equ, color="red")
plt.show()

Histogram Matching

Taken from PyImageSearch

Step 1: Calculate Histograms Compute the histograms of the source image (Hs) and target image (Ht) for intensity values (r).

Step 2: Calculate CDFs Compute the cumulative distribution functions (CDFs) for the source image (CDFs) and target image (CDFt).

Step 3: Establish Correspondence Find the corresponding intensity values between the source and target images using the inverse CDF of the target image.

Step 4: Apply Transformation Apply the intensity transformation to the source image using the established correspondence.

Step 5: Verify Similarity Calculate the mean absolute difference between the transformed source image and the target image to verify their similarity.

(np.float64(-0.5), np.float64(1199.5), np.float64(799.5), np.float64(-0.5))

Diferencia media absoluta: 4.501011458333333
# Cargar la imagen fuente y objetivo
img_s = cv2.imread('imagen_fuente.jpg')
img_t = cv2.imread('imagen_objetivo.jpg')

# Calcula los histogramas
hist_s = cv2.calcHist([img_s], [0], None, [256], [0, 256])
hist_t = cv2.calcHist([img_t], [0], None, [256], [0, 256])

# Calcula las CDF
cdf_s = np.cumsum(hist_s)
cdf_t = np.cumsum(hist_t)

# Establece la correspondencia
r_t = np.interp(cdf_s, cdf_t, np.arange(256))

# Aplica la transformación
img_t_match = cv2.LUT(img_s, r_t)

# Verifica la similitud
diff = np.mean(np.abs(img_t_match - img_t))

print(f'Diferencia media absoluta: {diff}')